Theorem topbas 12861

Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
topbas	⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Proof of Theorem topbas

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inopn 12795	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∩ 𝑦) ∈ 𝐽)
2	1	3expb 1199	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (𝑥 ∩ 𝑦) ∈ 𝐽)
3		simpr 109	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦))
4		ssid 3167	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
5	3, 4	jctir 311	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
6		eleq2 2234	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑥 ∩ 𝑦)))
7		sseq1 3170	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑤 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
8	6, 7	anbi12d 470	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))))
9	8	rspcev 2834	. . . . . 6 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
10	2, 5, 9	syl2an2r 590	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
11	10	exp31 362	. . . 4 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑥 ∩ 𝑦) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))))
12	11	ralrimdv 2549	. . 3 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽) → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
13	12	ralrimivv 2551	. 2 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
14		isbasis2g 12837	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
15	13, 14	mpbird 166	1 ⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449   ∩ cin 3120   ⊆ wss 3121  Topctop 12789  TopBasesctb 12834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797  df-top 12790  df-bases 12835

This theorem is referenced by:  resttop  12964  txtop  13054